JEE Mains Physics · Units and Measurements

Dimensional Homogeneity and Unknown Constants

Only quantities with the same dimensions can be added, subtracted or equated, and the argument of a sine, an exponential or a logarithm has no dimensions; these two rules fix the dimensions of every unknown constant in an equation.

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Why this matters

Twenty-six PYQs, all multiple choice, and two from 2026. Fourteen find constants from terms that are added together, six of them the a and b of the van der Waals equation; eight use the rule that the argument of a sine, an exponential or a logarithm has no dimensions; four test whether a whole equation is dimensionally correct. The method is the same every time: find each constant on its own, then combine.

Concept 1 of 3: Constants in terms that are added or subtracted

You cannot add a length to a time. So in any sum, every term has the dimensions of the quantity on the left, and a constant added to a variable inside a bracket has the dimensions of that variable. Use this term by term: each term gives one equation for one constant.

Definition

In $Q = X + Y$ , $[X] = [Y] = [Q]$ .
A constant added to a variable, as in $(t + c)$ or $(V - b)$ , has that variable's dimensions.
Van der Waals, $\left(P + \dfrac{a}{V^{2}}\right)(V - b) = RT$ : $[a/V^{2}] = [P]$ , so $[a] = ML^{5}T^{-2}$ ; $[b] = L^{3}$ .
Hence $a/b$ is an energy (it has the dimensions of PV), $a/b^{2}$ is a pressure or energy density, and $b^{2}/a$ is a compressibility.
Find each constant separately, then multiply or divide for the combination asked.

Principle of homogeneity

[X] = [Y] = [Q] \text{ in } Q = X + Y \qquad \left[\frac{a}{V^{2}}\right] = [P],\ [b] = [V]

Worked example

The position of a particle is

x = at^{3} + \dfrac{b}{t + c}

, where t is time. Find the dimensions of

\dfrac{ab}{c}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 21 Jan 2026 Shift 1 · Q7Moderate

Example 1 · Units and Measurements · Dimensional Homogeneity and Unknown Constants

Consider a modified Bernoulli equation.

\left( P +\frac{A}{Bt^{2}} \right)+ \rho g(\text{ }h + Bt) +\frac{1}{2}\rho{\text{ }V}^{2}=\text{~constant~}

If t has the dimension of time then the dimensions of A and B are

\_\_\_\_

\_\_\_\_

respectively.

The constant inside a bracket takes the variable's dimensions

In V − b, the constant b is a volume. It is not found from RT; it is fixed by the quantity it is subtracted from. The same holds for c in t + c, which is a time.

a/V² is a pressure, so a is not a pressure

The term a/V² has pressure's dimensions, so a itself is pressure × volume², ML⁵T⁻². Giving a the dimensions of pressure drops the V².

Drill 13 more on constants in terms that are added or subtracted

Concept 2 of 3: Arguments of sine, exponential and logarithm are dimensionless

A sine, an exponential or a logarithm can only act on a pure number. So whatever sits inside must have no dimensions, and the function itself has none either. The whole dimension of the term then rests on the factor in front.

Definition

In $\sin(Bx)$ , $e^{-Bt}$ or $\log(Bx)$ : $[Bx] = 1$ , so $[B] = [x]^{-1}$ .
The function's value is dimensionless, so in $Q = A\sin(Bx)$ , $[A] = [Q]$ .
In $e^{-\beta x^{2}/(kT)}$ , $\beta x^{2}$ has the dimensions of $kT$ , an energy.
In a wave $y = a\sin(\omega t - kx)$ , $[\omega] = T^{-1}$ , $[k] = L^{-1}$ , and $\omega/k$ is a speed.

Argument rule

Q = A\sin(Bx) \Rightarrow [B] = [x]^{-1},\ [A] = [Q]

Worked example

A pressure is given by

P = Ae^{-Bt} + C\log(Dx)

, where t is time and x is a length. Find the dimensions of

\dfrac{AB}{CD}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2021 · 2021 Compilation Paper 15 · Q2Moderate

Example 2 · Units and Measurements · Dimensional Homogeneity and Unknown Constants

For the force equation

F=Acos(Bx) +Csin(Dt)

. Find dimensions of

\frac{AD}{B}

The prefactor carries the whole dimension

In Q = A sin(Bx), the sine is a pure number, so A has exactly the dimensions of Q. Giving A some of B's dimensions, as if the sine passed them on, is the usual slip.

Drill 7 more on arguments of sine, exponential and logarithm are dimensionless

Concept 3 of 3: Checking whether an equation is dimensionally correct

Write the dimensions of each side, and of each added term, separately. If any two differ, the equation is wrong. If they all match, the equation may still be wrong: a missing 2 or π is invisible to dimensions. So the test can reject an equation but never prove it.

Definition

Both sides, and every added term, must have the same dimensions.
Pure numbers (2, π, ½) and dimensionless functions cannot be checked this way.
Kepler: $T^{2} = \dfrac{4\pi^{2}r^{3}}{GM}$ ; mass-energy: $E^{2} = p^{2}c^{2} + m^{2}c^{4}$ .
A dimensionally correct equation can still be physically wrong.

Homogeneity test

[\text{left side}] = [\text{right side}] = [\text{every added term}]

Worked example

The speed of a wave on a string of tension T and mass per unit length μ is claimed to be either

v = \sqrt{T/\mu}

v = \sqrt{T\mu}

. Which is dimensionally correct?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 4 April 2024 · Q109Moderate

Example 3 · Units and Measurements · Dimensional Homogeneity and Unknown Constants

Applying the principle of homogeneity of dimensions, determine which one is correct. where

T

is time period,

G

is gravitational constant,

M

is mass,

r

is radius of orbit.

Dimensionally correct does not mean correct

T = π√(l/g) passes the dimension test but is wrong by a factor 2. A dimension check can only rule an equation out.

Drill 3 more on checking whether an equation is dimensionally correct

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Constants in terms that are added or subtracted
Principle of homogeneity
$[X] = [Y] = [Q] \text{ in } Q = X + Y \qquad \left[\frac{a}{V^{2}}\right] = [P],\ [b] = [V]$
Arguments of sine, exponential and logarithm are dimensionless
Argument rule
$Q = A\sin(Bx) \Rightarrow [B] = [x]^{-1},\ [A] = [Q]$
Checking whether an equation is dimensionally correct
Homogeneity test
$[\text{left side}] = [\text{right side}] = [\text{every added term}]$

Watch out for (4)

The constant inside a bracket takes the variable's dimensions
→ Constants in terms that are added or subtracted
a/V² is a pressure, so a is not a pressure
→ Constants in terms that are added or subtracted
The prefactor carries the whole dimension
→ Arguments of sine, exponential and logarithm are dimensionless
Dimensionally correct does not mean correct
→ Checking whether an equation is dimensionally correct

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